Timeline for Most general definition of differentiation
Current License: CC BY-SA 4.0
13 events
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| Dec 6, 2019 at 13:12 | vote | accept | Ponta | ||
| Dec 5, 2019 at 10:55 | comment | added | Dave L Renfro | And to really get a sense of how deep this rabbit hole goes, see Derivation and Martingales by Hayes/Pauc, Differentiation of Integrals in R $^n$ by Guzman, and the infamous Web Derivatives by Kenyon/Morse. | |
| Dec 5, 2019 at 10:55 | comment | added | Dave L Renfro | A very nice survey of this work is given in Bruckner's Differentiation of Integrals and the more recent survey paper Differentiation by Thomson (Chapter 5, pp. 179-247, in Vol. I of Handbook of Measure Theory). (continued) | |
| Dec 5, 2019 at 10:33 | comment | added | Dave L Renfro | To complement the "real functions of a real variable" variations in "this answer" of mine you cited, and somewhat different from the various multivariable (including infinite-dimensional) variations you mentioned, there is a seemingly endless variety of notions based on work done in the 1930s through 1960s (mostly) on derivation bases that grew out of attempts to refine and generalize the Fundamental Theorem of Calculus by Busemann/Feller, Zygmund, Saks, Denjoy, Trjitzinsky, Haupt, Pauc, Morse, Hayes, Guzman, and others. (continued) | |
| Dec 5, 2019 at 9:05 | history | edited | Dirk | CC BY-SA 4.0 |
added 639 characters in body
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| Dec 5, 2019 at 0:39 | comment | added | user76284 | How about a partial order? | |
| Dec 4, 2019 at 17:41 | comment | added | Dirk | In some cases, some notion is stronger than another, but in other cases they are not comparable (e.g. the weak derivative is used for functions on real domains and the subgradient is defined for convex functions on normed spaces and even in the cases where both are applicable they do not really compare well). | |
| Dec 4, 2019 at 16:23 | comment | added | Andrew Tawfeek | Why couldn't they be ordered by generality? Can't we have that if X is a special case of Y, then Y $\geq$ X in terms of generality? | |
| Dec 4, 2019 at 15:09 | comment | added | Willie Wong | @DanieleTampieri: essentially the difference between Bouligand and Frechet (as described in jstor.org/stable/pdf/44001767.pdf) is that in Frechet differentiability, you assume the function is at $x$ approximated by a linear function, while in Bouligand you assume the function is at $x$ approximated by a function that is merely one-homogeneous. | |
| Dec 4, 2019 at 15:07 | comment | added | Willie Wong | @DanieleTampieri: according to ijpam.eu/contents/2013-83-3/7/7.pdf, given $X, Y$ normed vector spaces, a map $f:X\to Y$ is B-differentiable at $x$ if $\exists L$ such that $\| f(y) - f(x) \| \leq L \|y - x\|$ for every $y$ in a small open neighborhood of $x$, and for every vector $v$, the limit $\lim_{t \searrow 0} \frac{1}{t} (f(x + tv) - f(x) )$ exists. | |
| Dec 4, 2019 at 15:01 | history | edited | Willie Wong | CC BY-SA 4.0 |
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| Dec 4, 2019 at 13:45 | comment | added | Daniele Tampieri | I'd like to have some information on Bouligand differentiability: I googled the sentence but I found only papers related to specific problems, nothing introductory. Do you have some suggestion on where to start approaching this topic? | |
| Dec 4, 2019 at 11:33 | history | answered | Dirk | CC BY-SA 4.0 |